DISCRETE-TIME ADAPTIVE LQG/LTR CONTROL

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1 DISCRETE-TIME ADAPTIVE LQG/LTR CONTROL DARIUSZ HORLA Poznan University of Technology Inst. of Control and Inf. Eng. ul. Piotrowo 3a, Poznan POLAND ANDRZEJ KRÓLIKOWSKI Poznan University of Technology Inst. of Control and Inf. Eng. ul. Piotrowo 3a, Poznan POLAND Abstract: An adaptive discrete-time LQG control with loop transfer recovery (LTR) is considered. The control problem is analyzed using state-space model and the parameter estimation problem is implemented for corresponding discrete-time model in transfer function form obtained via ZOH. Thus the direct estimation of model parameters is possible by means of standard RLS or ERLS procedure and the adaptive control is implemented through certainty equivalence principle. Computer simulations of third-order systems modeled by a second-order minimum-phase and nonminimum-phase models are given to illustrate the robustness and performance properties of the adaptive LQG/LTR controller, particularly with respect to the weighting parameter. Key Words: LQG control. Loop transfer recovery. Adaptive control Introduction The problem of adaptive LQG control is not much investigated in the literature, in particular this holds for adaptive LQG control with loop transfer recovery (LTR). Adaptive LQG control has been discussed e.g. in [3,, 2, 3, 4, 4], where in [2] an adaptive LQG/LTR problem was solved augmenting the basic estimator-based controller with a stable proper linear system feeding back the estimation residuals. This idea was also used for non-adaptive continuous-time systems in [] using the H /H 2 optimization technique. The key issue in adaptive LQG control is the closedloop identifiability. For example in [3] a discretetime system and a cost-biased least-squares parameter estimation was used in order to achieve overall asymptotic system optimality. In [4] a continuous-time system was considered and a modified weighted leastsquares parameter estimation algorithm was used to obtain good properties of estimates. In both papers only a fully state observation case was considered. In this paper, LQG adaptive control with posssible application of LTR technique is considered. The adaptive discrete-time LQG control algorithm is proposed where the controller/filter parameters are tuned on the basis of transfer function model identification. Asymptotic performance and robustness properties are analyzed and simulations for third-order system considered as a second-order model are given. The role of tuning parameter is underlined. 2 LTR for Discrete-Time System The discrete-time system obtained with ZOH is given by x t+ = Fx t + Gu t + w t () y t = Hx t + v t (2) where {w t } and {v t } are sequences of independent random vector variables with zero mean and covariances Ew t w T s = Σ wδ t,s, v t v T s = Σ vδ t,s. The Kalman predictor in steady-state is given by ˆx t+/t = F ˆx t/t + Gu t + K p ỹ p t (3) where ỹ p t = y t Hˆx t/t is an innovation of output at time t. The predictor gain is given by K p = FP f H T [HP f H T + Σ v ] (4) where P f is the solution of Riccati equation P f = FP f F T +Σ w FP f H T [HP f H T +Σ v ] HP f F T (5) The covariance of the innovation ỹ p is Σỹ = t HP f H T + Σ v. Filtered estimate ˆx t/t in terms of ˆx t/t is ˆx t/t = ˆx t/t + K f ỹ p (6) t and its recursive version is ˆx t+/t+ = F ˆx t/t + (I K f H)Gu t + K f ỹ f t+ (7) where ỹ f t+ = y t+ HF ˆx t/t and the filter gain K f = P f H T [HP f H T + Σ v ], (8) ISBN:

2 so K p = FK f in view of (4). An alternative equation for (7) is ˆx t+/t+ = F ˆx t/t + Gu t + K f ỹ p t+ The LQG control law (9) u t = K c x t/t () aims to minimize the performance criterion where J = E x T t Qx t + u T t Ru t. () t= K c = [G T P c G + R] G T P c F (2) and P c is the solution of Riccati equation P c = F T P c F F T P c G[G T P c G+R] G T P c F +Q (3) It is worthy noting that the optimal performance is J opt = tr[p c Σ w +P c G(G T P c G+R) G T P c FP f F T ]. (4) When weighting matrices Q and R in the performance criterion () take form Q = H T H, R = I and det(hg) (ie.the system is square) then asymptotically as, one have the case of the simple control with criterion J = E y T t y t, (5) t= and the criterion (4) taking a simple form J opt = trh T H[Σ w + FP f F T ]. (6) Moreover, assuming that the system (), (2) is stabilizable and detectable it can be shown [7], [5] that K c takes then very simple form K c = (HG) HF. (7) If G(z) = H(zI F) G is minimum-phase (mph) and K c takes a form (7) then the perfect recovery takes place, that is for and the filter s open-loop return ratio is Φ(z) = H(zI F) K p. (2) Putting (7) into (8) it can be seen that (z) = so the recovery takes place. When G(z) is nonminimum-phase (nmph) then the perfect recovery is in general not possible, however is recommended and the possibility of recovery is frequently achieved in closed-loop bandwidth. The robustness can be measured by means of the H norm of sensitivity transfer function S(z) = (I + G(z)G f (z)). (2) For the Kalman predictor feedback, the controller is and its transfer function is u t = K c x t/t (22) G p (z) = K c [zi F + GK c + K p H] K p. (23) In this case the perfect recovery cannot be in general obtained. 3 Adaptive Control The SISO ARMAX model is given by A(q )y t = B(q )u t + C(q )e t (24) where A(q ),B(q ) and C(q ) are polynomials in the backward shift operator q, i.e. A(q ) = + a q a n q n,b(q ) = b q b n q n,c(q ) = + c q c n q n and y t is the output, u t is the control input, and {e t } is assumed to be a sequence of independent variables with zero mean and variance σe 2. Unknown system parameters θ = (a,...,a n,b,...,b n,c,...,c n ) T are estimated on-line to obtain an updated model at time t, i.e. ˆθ t. ARMAX model (24) has an equivalent innovation state space representation x t+ = Fx t + gu t + k e e t (25) y t = h T x t + e t (26) (z) = G(z)G f (z) Φ(z) =, (8) where g = (b,...,b n ) T, k e = (c a,...,c n a n ) T, h T = (,,...,) where the transfer function G f (z) of compensator defined by (7) and () can be manipulated into the form a... G f (z) = zk c [zi (I K f H)(F + GK c )]..... K f = F = a n..... = zk c [zi F GK c ] K f, (9) a n.... ISBN:

3 Kalman predictor (3) associated with eq.(25) is ˆx t+/t = F ˆx t/t + gu t + k p ỹ p t (27) where ỹ p t = y t h T ˆx t/t and σỹ,p 2 is the variance of ỹ p t for which it holds σ2 ỹ,p = σ2 e. The predictor gain is now given by k p = (FP f h + σ 2 ek e )[h T P f h + σ 2 e] (28) where P f is the solution of Riccati equation P f = FP f F T + k e k T e σ 2 e (FP f h + k e σ 2 e) [h T P f h + σ 2 e ] (FP f h + k e σ 2 e )T (29) The actual model used for LQG/LTR control is obtained for current parameter estimates ˆθ t. The LTR control law (7) is especially useful for adaptive control because it does not need solving the Riccati equation (3) for every ˆθ t, and the feedback gain K c can be tuned directly. The investigated problem is to check out how the design parameter used in the LTR technique can influence the performance of adaptive control in case of undermodeling. It is supposed that used in adaptive LQG/LTR control can be tuned to compromize the LQG/LTR control performance, robustness and parameter estimation quality. The issue of stability of the proposed adaptive LQG/LTR control system is of course crucial. This depends not only on the magnitude of modeling error but also on the asymptotic convergence of parameter estimates. Particularly, one should take into account that in general the parameter estimation in LQG adaptive control even in the lack of modelling error, does not assure the convergence to the true parameters. Closed loop stability and good performance cannot be guaranteed especially during the transient stage. 4 Simulations Consider an example of a third-order actual system G (s) = s + 2 (s + )(s + 3) + η s + 2 whose nominal model G(s) is mph, so the case η = corresponds to the lack of modeling error and η = is the case of undermodeling. Discretizing the continuous-time system with ZOH and sampling period T s =.5s yields the following transfer function in q operator G (q ) =.3262q.224q q η.3q.3679q. (3) As already mentioned, a second-order model was taken for identification and certainty equivalence principle was used to implement the adaptive control system to demonstrate the robustness of adaptive LQG/LTR controller with respect to undermodeling. The corresponding RLS parameter estimates of the nominal model are shown in Fig., for = and =.5. Fig.2 shows the case of undermodeling (η = ). From Fig. one can see that for η = parameter estimates converge to the true parameters of second-order nominal model. The convergence holds for any value of, however the bigger the value of the slower the estimation convergence and the worse performance of adaptive control can be expected. It is also to observe that when η = the parameter estimates converge to some stationary points, however different from the true ones and in general different for various. In this situation a more significant deterioration of control criterion is to be expected. This is illustrated in Fig.3 where the simulated criterion () is plotted versus for known parameters and for adaptive control with η = and η =. The criterion values for = are J =.6,.693,.975, respectively. As a second example consider a third-order actual system G (s) = s + (s + )(s + 2) + η s + 3 whose nominal model is nmph. Discretizing the continuous-time system with ZOH and sampling period T s =.5s yields the following transfer function in q operator G (q ) =.62q q q +.223q 2 +η.259q.223q. (3) From Fig.4 one can see that for η = parameter estimates again converge to the true parameters of second-order nominal model for all. The case with η =.5 is shown in Fig.5. This time, again the parameter estimates converge to some stationary points, however different in general for various. Fig.6 represents a simulated criterion () plotted versus for known parameters and adaptive control of nmph nominal model with η = and η =.5. In case of greater values of η the adaptive system gets unstable while for nominal system the performance is quite good. The criterion values for = are J =.559,.696,2.7926, respectively. The norms S(z), G(z)G f (z) and G δ (z)g f (z)) for known parameters are plotted as shown in Fig.7 for η = where G δ (z) is an additive perturbation, i.e. G = G + ηg δ. It can ISBN:

4 be observed that does influence the sensitivity and stability for mph system is assured for all. For considered example of nmph system we can see that the system gets unstable for <.26. Obviously, estimation procedure introduces an additional lack of modeling in the form of uncertain estimates that can worsen stability robustness. System parameters were identified using the RLS algorithm and output noise variance σe 2 was set at.. Figures show that for both mph and nmph nominal systems the smaller the weight the better the estimates convergence rate where this effect is stronger in the case of η = when the estimates tend to the true nominal values. Obviously, in the case of more general models, the recursive pseudolinear regression (RPLR) or recursive prediction error (RPEM) algorithms should then be applied. The results shown in [8], confirm that RPEM is more suitable in the considered undermodelled situation taking into account the asymptotic properties of the algorithm..5 = J Recent Advances in Telecommunications and Circuits 5 CONCLUSIONS Application of loop transfer recovery technique in the context of adaptive discrete-time LQG control is presented. Parameter estimation of ARMAX model transfer function is used for tuning the discrete-time LQG/LTR compensator. The interplay between robustness, performance and estimation convergence with respect to the weight is underlined, particularly when =. Examples of third-order actual systems known parameters, η = adaptive system, η = adaptive system, η = Figure 3: MPH: plot of J versus.5 a b =.5.5 = Figure : MPH: estimates for η = = Figure 4: NMPH: estimates for η =.5 =.5 a a 2.5 b b 2 = = = Figure 2: MPH: estimates for η = Figure 5: NMPH: estimates for η =.5 ISBN:

5 J Recent Advances in Telecommunications and Circuits described by a second-order mph and nmph nominal models are taken for simulation. Simulation results show an effectivness of the adaptive LQG control with possibility of using the LTR tuning parameter as a way for robustifying the adaptive control. References: [] R.R. Bitmead and M. Gevers and V. Wertz, Adaptive Optimal Control, Prentice Hall International 99. [2] T.T. Tay and J.B. Moore, Adaptive LQG controller with loop transfer recovery, Int.J.Adapt.Contr.Sign.Proc., 99, Vol. 5(2), pp [3] A. Krolikowski, Amplitude constrained adaptive LQG control of first order systems, Int.J.Adapt.Contr.Sign.Proc., 995, Vol. 9(3), pp [4] P.M. Mäkilä and T. Westerlund and H.T. Toivonen, Constrained linear quadratic gaussian control with process applications, Automatica, 984, Vol. 2(), pp known parameters, η = adaptive system, η = adaptive system, η =.5 [5] J.M. Maciejowski, Asymptotic recovery for discrete-time systems, IEEE Trans. Automat. Contr., 985, Vol. 3(6), pp [6] J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley 989. [7] M. Tadjine, and M. M Saad and L. Dugard, Discrete-time compensators with loop transfer recovery, IEEE Trans. Automat. Contr., 994, Vol. 39(6), pp [8] M. Nilsson and B. Egardt, Comparing recursive estimators in the presence of unmodeled dynamics, Proc. IFAC Symp. ALCOSP, 2, pp , Antalya, Turkey. [9] M. Athans, A tutorial on the LQG/LTR method, Proc. American Control Conf, 986, LIDS P 542, Seattle, USA. [] T.T. Tay and J.B. Moore, Loop recovery via H /H 2 sensitivity recovery, Int.J.Control, 989, Vol. 49(4), pp [] B. Kulcsar, LQG/LTR controller design for an aircraft model, Periodica Polytechnica, Ser.Transp.Eng, 2, Vol. 28( 2), pp [2] I.Kavanbod-Hosseini, and D. Noll, and P.Apkarian, On a generalization of the LTR procedure, Proc. Control Decision Conf, 2, CD ROM 542, Atlanta, USA. [3] P.R. Kumar, Optimal adaptive control of linearquadratic-gaussian systems, SIAM J. Control and Optimization, 983, Vol. 2(2), pp [4] T.E. Duncan, and B.Pasik-Duncan, Adaptive continuous-time linear quadratic gaussian control, IEEE Trans. Automat. Contr., 999, Vol. 44(9), pp Figure 6: NMPH: plot of J versus.25 MPH.8 NMPH.2.6 S.5..5 S GG f.4.3 GG f G f G p S.2.5 G f G p S Figure 7: H norms versus ISBN: